Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3019 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#120 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | C2×#379 |
| prime radical | C2×#585 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1824 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,10,5,18)(3,12,14,8,15,23,19,22)(4,9,16,26,17,24,20,25)(13,28,31,32,30,27,29,21);(1,3)(2,8)(4,13)(5,19)(6,14)(7,15)(9,21)(10,22)(11,23)(12,18)(16,31)(17,30)(20,29)(24,32)(25,27)(26,28);(1,4)(2,9)(3,13)(5,20)(6,16)(7,17)(8,21)(10,24)(11,26)(12,27)(14,31)(15,30)(18,25)(19,29)(22,32)(23,28);(2,10)(8,22)(9,24)(11,18)(12,23)(21,32)(25,26)(27,28)Polycyclic presentation:
2^8|a^2=g,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,e^b=eh,f^b=fh,e^c=eh58 conjugacy classes — view the full character table.
| maximal subgroup of | #147753, #147754, #147758, … |
| maximal normal subgroup of | #147753, #147754, #147758, … |
| maximal quotient of | #184930, #184931, #184932, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13608);